Difference between revisions of "Geometric distribution/Infobox"

	Geometric Distribution
	X∼Geo(p); for p the probability of each trials' success
	X=k means that the first success occurred on the kth trial, k∈N≥1
Definition
Defined over	X may take values in N≥1={1,2,…}
p.m.f	P[X=k]:=(1−p)k−1p
c.d.f / c.m.f	P[X≤k]=1−(1−p)k
cor:	P[X≥k]=(1−p)k−1
Properties
Expectation:	E[X]=1p
Variance:	Var(X)=1−pp2

Latest revision as of 15:13, 16 January 2018

PAGE FOR TRANSCLUSION, YOU PROBABLY WANT GEOMETRIC DISTRIBUTION

@@ Line 22: / Line 22: @@
 |data10={{MM|\mathbb{E}[X]\eq\frac{1}{p} }}<ref>See ''[[Expectation of the geometric distribution]]''</ref>
 |label11=[[Variance]]:
-|data11={{Nowrap|{{XXX|Unknown}}<ref group="Note">Due to different conventions on the definition of geometric (for example {{M|X':\eq X-1}} for my {{M|X}} and another's {{M|X'\sim\text{Geo}(p)}}) or even differing by using {{M|1-p}} in place of {{M|p}} in the {{M|X}} and {{M|X'}} just mentioned - I cannot be sure without working it out that it's {{MM|\frac{1-p}{p^2} }} - I record this value only for a record of what was once there with the correct expectation - DO NOT USE THIS EXPRESSION</ref>}}
+|data11={{Nowrap|{{MM|\text{Var}(X)\eq\frac{1-p}{p^2} }}<ref>See ''[[Variance of the geometric distribution]]''</ref>}}
 }}<noinclude>
 : '''PAGE FOR TRANSCLUSION, YOU PROBABLY WANT [[Geometric distribution|''GEOMETRIC DISTRIBUTION'']]'''

Definition
Geometric Distribution
$X\sim\text{Geo}(p)$ for $p$ the probability of each trials' success
$X\eq k$ means that the first success occurred on the $k^\text{th}$ trial, $k\in\mathbb{N}_{\ge 1}$
Defined over	$X$ may take values in $\mathbb{N}_{\ge 1}\eq\{1,2,\ldots\}$
p.m.f	$\mathbb{P}[X\eq k]:\eq (1-p)^{k-1}p$
c.d.f / c.m.f^{[Note 1]}	$\mathbb{P}[X\le k]\eq 1-(1-p)^k$
cor:	$\mathbb{P}[X\ge k]\eq (1-p)^{k-1}$
Properties
Expectation:	$\mathbb{E}[X]\eq\frac{1}{p}$ ^[1]
Variance:	$\text{Var}(X)\eq\frac{1-p}{p^2}$ ^[2]